Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{a^3 + 2a^2 - 24a}{a^3 + 12a^2 + 36a}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {a(a^2 + 2a - 24)} {a(a^2 + 12a + 36)} $ $ y = \dfrac{a}{a} \cdot \dfrac{a^2 + 2a - 24}{a^2 + 12a + 36} $ Simplify: $ y = \dfrac{a^2 + 2a - 24}{a^2 + 12a + 36}$ Since we are dividing by $a$ , we must remember that $a \neq 0$ Next factor the numerator and denominator. $ y = \dfrac{(a + 6)(a - 4)}{(a + 6)(a + 6)}$ Assuming $a \neq -6$ , we can cancel the $a + 6$ $ y = \dfrac{a - 4}{a + 6}$ Therefore: $ y = \dfrac{ a - 4 }{ a + 6 }$, $a \neq -6$, $a \neq 0$